Internal stabilization of Navier-Stokes equations with finite-dimensional controllers

Barbu, Viorel; Triggiani, Roberto

doi:10.1512/iumj.2004.53.2445

Cited by 133 publications

(169 citation statements)

References 10 publications

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“…Let us note that a similar result was proved earlier in [BRS11] for the 3D Navier-Stokes system (see also [Fur04,Bar03,BT04] for some results on stabilisation to a stationary solution). Here we establish some additional properties for the control.…”

Section: Introductionsupporting

confidence: 78%

Control and mixing for 2D Navier-Stokes equations with space-time localised noise

Shirikyan¹

2015

Ann. Sci. École Norm. Sup.

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We consider randomly forced 2D Navier-Stokes equations in a bounded domain with smooth boundary. It is assumed that the random perturbation is non-degenerate, and its law is periodic in time and has a support localised with respect to space and time. Concerning the unperturbed problem, we assume that it is approximately controllable in infinite time by an external force whose support is included in that of the random force. Under these hypotheses, we prove that the Markov process generated by the restriction of solutions to the instants of time proportional to the period possesses a unique stationary distribution, which is exponentially mixing. The proof is based on a coupling argument, a local controllability property of the Navier-Stokes system, an estimate for the total variation distance between a measure and its image under a smooth mapping, and some classical results from the theory of optimal transport.

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Section: Introductionsupporting

confidence: 78%

Control and mixing for 2D Navier-Stokes equations with space-time localised noise

Shirikyan¹

2015

Ann. Sci. École Norm. Sup.

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“…One way to construct robust feedback laws consists in using the methods of the optimal control theory. This approach has been studied in the case of an internal control [2,1,3], and has been numerically tested with a boundary control in the very specific geometry of the rectangular driven cavity [10] and when the normal component of the control is equal to zero. In many engineering applications [13,11] the normal component of the control variable is not equal to zero.…”

Section: Introductionmentioning

confidence: 99%

Local Boundary Feedback Stabilization of the Navier-Stokes Equations

Raymond¹

2006

Proceedings of Control Systems: Theory, Numerics and Applications — PoS(CSTNA2005)

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We study the exponential stabilization of the linearized Navier-Stokes equations around an unstable stationary solution, by means of a feedback boundary control, in dimension 2 or 3. The feedback law is determined by solving a Linear-Quadratic control problem. We do not assume that the normal component of the control is equal to zero. In that case the state equation, satisfied by the velocity field y, is decoupled into an evolution equation satisfied by Py, where P is the so-called Helmholtz projection operator, and a quasi-stationary elliptic equation satisfied by (I − P)y. Using this decomposition we show that the feedback law can be expressed only in function of Py. We show that the linear feedback law provides a local exponential stabilization of the Navier-Stokes equations.

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“…From Theorem 2.1 on page 1448 of [2], we know that for all y 0 ∈ H, there exists a control u y 0 depending on y 0 so that…”

Section: Results From Other Sourcesmentioning

confidence: 99%

Min-Max Game Theory for the Linearized Navier-Stokes Equations with Internal Localized Control and Distributed Disturbance

Spencer¹

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Internal stabilization of Navier-Stokes equations with finite-dimensional controllers

Cited by 133 publications

References 10 publications

Control and mixing for 2D Navier-Stokes equations with space-time localised noise

Control and mixing for 2D Navier-Stokes equations with space-time localised noise

Local Boundary Feedback Stabilization of the Navier-Stokes Equations

Min-Max Game Theory for the Linearized Navier-Stokes Equations with Internal Localized Control and Distributed Disturbance

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